Impulsive Differential Equations

Periodic Solutions and Applications

Drumi Bainov, Pavel Simeonov

• Publisher: Routledge
• Year: 2017
• Print ISBN: 9780367845544
• eBook ISBN: 9781351439091
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Impulsive differential equations have been the subject of intense investigation in the last 10-20 years, due to the wide possibilities for their application in numerous fields of science and technology. This new work presents a systematic exposition of the results solving all of the more important problems in this field.

Chapter I

Preliminary Notes

In this chapter we shall formulate some auxiliary assertions about impulsive differential equations; we shall use these ideas in the subsequent chapters.

1. Impulsive differential equations

Before we describe impulsive differential equations we shall consider two examples in order to show that the impulses appear in a natural way in the investigation of concrete processes.

Example 1.1 Consider the scalar differential equation of Verhulst:

$\frac{\text{d}x}{\text{d}t}=rx\left(1-\frac{x}{K}\right).$

(1.1)

Equation (1.1) describes the change of the population number x(t) of an isolated population of some biological species with incessant reproduction inhabiting a stationary environment with limited resources. The constants r > 0 and K > 0 are called the rate coefficient and the capacity of the environment, respectively.

The forms of the graphs of the positive solutions of equation (1.1) are given in Fig. 1.1. For the sake of definiteness we shall assume that x(t) is the quantity of biomass of a certain species of micro-organism cultivated in a bioreactor. External effects on the development of the species can cause jumps in the quantity of biomass x(t). For instance, this is possible with a single removal of part of th...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Notation
• Chapter I: Preliminary Notes
• Chapter II: Linear Impulsive Periodic Equations
• Chapter III: Method of the Small Parameter. Non-critical Case
• Chapter IV: Method of the Small Parameter. Critical Case
• Chapter V: Non-linear Periodic Impulsive Equations
• Chapter VI: Approximate Analytical Methods for Finding Periodic Solutions
• References
• Author index
• Subject index